Claude account creation turned out to work with a domain I have hosted on Dreamhost. Although Dreamhost works, emails to my domains hosted on shared hosting at Hostgator are never replied to by their verification system.

Anthropic's support chatbot is happy to talk to me using a Hostgator email address, though. It says it doesn't have access to block information.

Somewhere Anthropic probably has a blacklist related to something on Hostgator shared hosting. Maybe I'll get my sites there moved to a different server.

If the concern is business practices, privacy, etc. adding Google to the mix doesn’t seem great

Or use SimpleLogin.